Let f V W be an isomorphism it a rule that is onetoone onto

Let f: V --> W be an isomorphism (i.t. a rule that is one-to-one, onto and preserves additon and salign in the sense that f(v+w) = f(v) + f(w) and f(cv) = cf(v)). Suppose that S = {v_1...,v_n} is a subset of V and let f(S) = {f(v_1),...,f(v_n)} be the associated subset of W.

a. Show that if S spans V, then f(S) spans W.

b. Show that if S is linearly independent then f(S) is linearly independent.

c. Deduce that if S is a basis of V, then f(S) is a basis of W.

Can someone please walk me through the steps of each of these so that I can understand where to start?

Thank you.

Solution

a.

f is onto so for any w in W there exist u in V so that:f(u)=w

S spans V so for any u in V there exist:

a1,...,an so that:

u=a1v1+...anvn

f(u)=z=f(a1v1+...+anvn)=a1f(v1)+...+anf(vn)

Hence, f(v1),.... f(vn) spans W since w in W was arbitrary

b.

Let S be linearly independent

Let, a1,....,an so that

a1f(v1)+...anf(vn)=0

f(a1v1+...+anvn)=0

Since, f is injective hence,

a1v1+....+anvn=0

But v1,...,vn are linearly independent so:a1=....=an=0

c.

Let S be a basis of V

Hence S spans V. From part (a) f(S) spans W

S is basis implies S is a linearly independent set. From part (b) f(S) is a linearly independent set

Hence, f(S) is a linearly independent set that spans W

Hence, f(S) is a basis for W